# Mistakes give rise to Problems- 16

Probability Level 5

You can remove the brackets in a fraction (note that this is fraction, not the Legendre symbol) , just like in the following fraction $\displaystyle \biggl(\color{#3D99F6}{\dfrac{a}{b}} \biggr) = \color{#3D99F6}{\dfrac{a}{b}}$

But if you do that in the $\color{#69047E}{\textbf{Binomial Coefficient}}$, like $\displaystyle \dbinom{a}{b} = \dfrac{a}{b}$ then it's a $\color{#D61F06}{\textbf{Big Mistake !!!}}$

But for how many ordered pairs $\color{#20A900}{(a,b)}$ such that $0\leq a \leq 20$ and $0\leq b \leq 20$ is the above said "false" property seen to be "true" ?

Details and assumptions

$\bullet\quad$ Factorial ($a!$) is defined only for non-negative integers.

$\bullet\quad$$\dbinom{a}{b} = \dfrac{a!}{b! \times (a-b)!}$

$\bullet\quad$$\dbinom{a}{b} = 0$ if $a.

$\bullet\quad$$0! =1$

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